Jack had taken as much time as he'd let himself with Jason, but there was one more thing he needed to do before heading out. Carefully, cautiously, he re-opened the place in the back of his mind that linked to Brant. Not enough to call out, just enough to be aware of his presence, and for Brant to know he was there. Not enough to let Brant know (hopefully) what was going on inside his head.
Then, he showered again, ate again, drank more water, put on fresh clothes that still didn't involve his coat because he was disheartened every time he felt how big it was on his wasted frame, and went out in search of the Doctor. The TARDIS was a bit of a clue, but he wasn't confident enough to use his key, stopping outside instead to rap a rhythmic knock on the closed doors.
And belatedly hoped that River wasn't around.